Wednesday, February 28, 2018
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Jacques Distler and three co-authors (CDTZ, Austin and Maryland) have published an impressively technical paper "classifying" certain exotic beasts,

Tinkertoys for the \(E_8\) Theory.

Their paper is accompanied by a cool interactive website which you should investigate. In particular, if you're asking why they didn't include "all the stuff" from the website in their paper, try this subpage on four punctured spheres. Pick a combination of the four parameters and press "go". What you get are some diagrams of S-duality frames (1,025,438 of them) where you see two circles with some objects and left-right arrows connecting them. You can click at the objects and get additional data on the "fixtures".

If you don't understand what the data means and how to use them, don't worry. 7 billion people don't understand it and my estimate is that 30 people in the world may deal with the data – to the extent that they would have a chance to discover a typo if one were artificially introduced. ;-)

What did they do? They discussed some 4-dimensional quantum field theories with enhanced supersymmetry – the \(\NNN=2\) gauge theories in \(D=4\) – that may be derived from a master theory, the six-dimensional \((2,0)\) theory – by operations that require lots of group theory involving \(E_8\), the largest exceptional Lie group (that the laymen surely know as the Garrett Lisi group after a famous crackpot surfer), as well as geometric operations introducing and curing singularities within a two-sphere (injured by 3 or 4 punctures).

Now, all this science may be formally considered "non-gravitational quantum field theory with lots of geometry and group theory". So there's no full-blown string/M-theory in it. On the other hand, everything about their construction is completely stringy – in the sense that basically everyone who understands these quantum field theories and operations with them is a string theorist.

They're elaborating upon the "class-S" construction by Davide Gaiotto from April 2009 as well as Gaiotto, Moore, and Neitzke from July 2009. Now, I know Jacques Distler in person – as well as from tons of interactions on the web. I've spent years next to Davide Gaiotto and Andy Neitzke (who co-authored papers with AN) at Harvard, and years next to Greg Moore at Rutgers. So all this stuff has been pushed largely by brilliant minds whom I know very well. I believe that their expertise is rare and I think it's nontrivial to find grad students who would be able and willing to learn the necessary stuff even at the level of a "useful brute force assistant".

The six-dimensional \((2,0)\) theory is a non-gravitational local quantum field theory in 5+1 dimensions (well, a family of such theories). So formally, it's as non-gravitational or non-stringy as you can get. However, almost all its definitions and relationships that are known are tightly incorporated into the wisdom of string/M-theory. Note that \((2,0)\) refers to the amount of supersymmetry. In 5+1 dimensions, the supercharges transform as 5+1-dimensional spinors. Those spinors of \(Spin(5,1)\) include two inequivalent chiral representations, just like in \(Spin(4)\approx SU(2)\times SU(2)\) – which has the same types of spinors and where we "cancelled" the temporal and spatial dimensions – so unlike the case of \(Spin(3,1)\) or \(D=4\), it's not enough to say "how much SUSY". You must say "how much left-handed SUSY" and "how much right-handed SUSY". The \((2,0)\) number means that it's enhanced supersymmetry but both supersymmetries have the same chirality in six dimensions.

There are numerous relations of this field theory to string/M-theory. It may be obtained as a low-energy limit of M5-brane dynamics within M-theory; or as some approximation of type IIB string theory on a singularity within a 4-dimensional Euclidean space. It follows that the large \(N\) limit of the \(SU(N)\) \((2,0)\) theories is Maldacena-dual to a \(AdS_7\times S^4\) compactification of M-theory. That's the main link to the higher-dimensional physics.

But the \((2,0)\) has lots of cool relationships to lower-dimensional (compared to six) physics. It has lots of descendants. The \((2,0)\) theory may be formally associated with an ADE gauge group – either \(SU(N)\) or \(SO(2N)\) or \(E_6,E_7,E_8\). The first two are infinite families, the latter three are isolated exceptions. Distler et al. studied the largest isolated exception.

The very existence of these theories – which is "somewhat" hypothetical and if you deny the existence of string/M-theory, you could probably consistently deny the very existence of the \((2,0)\) theories as well – is enough to prove many marvelous facts about quantum field theories and string theory vacua. In particular, the \(\NNN=4\) gauge theory in \(D=4\) has the \(SL(2,\ZZ)\) S-duality group. This duality is obvious if you realize that the \(D=4\) theory may be obtained by compactifying the \((2,0)\) theory on a 2-torus: the duality is nothing else than the group of "large" coordinate transformations on the two-torus.

Distler et al., and the predecessors, consider a more complicated compactification than the simple two-torus. One replaces the two-torus by a more general Riemann surface, especially a simple two-sphere, one that may have "punctures", and there may be a "partial topological twist" associated with every "puncture". At any rate, there are many ways how to get rid of the two extra dimensions and dimensionally reduce the six-dimensional theory to a four-dimensional one.

One ends up with many \(\NNN=2\) theories in \(D=4\) – the "Seiberg-Witten" amount of supersymmetry in a four-dimensional spacetime. Depending on the types of topological twists etc., one gets many possible "descendants" of the six-dimensional theory. Some of them work, some of them don't work.

I can't explain all the details – one reason is that I obviously fail to understand all the details – but I want to make a general big-picture claim that is too philosophical that it doesn't appear in such papers, and maybe the authors aren't sufficiently playful to invent it.

As I announced in the subtitle, this construction of the four-dimensional theories follows the logic of Darwin's evolution – with common ancestors and mutations – while the usual bottom-up construction of quantum field theories (construction where we list possible pieces, fields, and combine them and their interaction) is the creationist attitude to build quantum field theories.

If the species were identified with four-dimensional quantum field theories, they could arise in two basic ways: creation and evolution. Creation means that God asks His assistant for the building blocks and decides, as if he were an engineer, how to combine the building blocks into a four-dimensional theory with some fields and interactions. So God decides it would be fun to have something like an elephant, so He asks his assistant for a trunk and just constructs an elephant.

Gaiotto, Neitzke, Distler, Moore, and others are approaching the birth of the four-dimensional theories differently, by the evolutionary path. They pick their ancestor, in this case the \(E_8\) \((2,0)\) theory in six dimensions, and let it live. They pick several clones of it and mutate them – which happens in the ordinary life – and they get various ancestors with a modified DNA.

A big difference is that in the creationist attitudes to the spectrum of the quantum field theories, you basically impose your conditions "how the theory should behave at long distances" as the initial constraints that dictate everything. On the other hand, the stringy evolutionary attitude is different. You allow the choices that may occur at the fundamental, DNA level – like the partial topological twists assigned to the punctures – and what the theory looks like at low energies is a surprise.

Note that this is exactly how it works in the real-world biological evolution. Some DNA may be mutated – DNA is more fundamental than the anatomy of an animal – and what the anatomy and physiology of the mutated animal arises is a surprise. Whether such a new animal species is "viable" is only determined later, by natural selection.

The stringy attitude based on the \((2,0)\) theory simply acknowledges that analogously to the DNA, some local choices in the Riemann surface of the 2 compactified dimensions are more fundamental, and the long-distance behavior of the resulting four-dimensional quantum field theory is an emerged, derived consequence of the more fundamental choices.

In biology, even if we didn't know about DNA, it would be more right to assume that "something like that exists" and is more fundamental (the more primary cause) than the anatomy of the animal species. This DNA-centered attitude is analogous to the stringy one and those who want to avoid the stringy origin of the theories are analogous to those who insist on creationism. Both of them imagine that it's scientifically OK to imagine that the Creator starts with the long-distance or anatomic result He wants to get – while the stringy or evolutionary folks know that the evolution of species or four-dimensional QFTs is a scientific process that has to obey some laws of Nature, too.